SELF-REPRESENTATIONS OF THE MÖBIUS GROUP NICOLAS MONOD AND PIERRE PY∗ Abstract. Contrary to the finite-dimensional case, the Möbius group admits interesting self-representations when infinite-dimensional. We construct and classify all these self-representations. The proofs are obtained in the equivalent setting of isometries of Lobachev- sky spaces and use kernels of hyperbolic type, in analogy to the classical con- cepts of kernels of positive and negative type. 1. Introduction 1.A. Context. For an ordinary connected Lie group, the study of its continuous self-representations is trivial in the following sense: every injective self-represen- tation is onto, and hence an automorphism. In the infinite-dimensional case, another type of “tautological” self-representa- tions presents itself. Namely, the group will typically contain isomorphic copies of itself as natural proper subgroups. For instance, a Hilbert space will be isomor- phic to most of its subspaces. Remarkably, some infinite-dimensional groups admit also completely differ- ent self-representations which are not in any sense smaller tautological copies of themselves. This phenomenon has no analogue in finite dimensions and the simplest case is as follows. ⋊ O Let E be a Hilbert space and Is(E) ∼= E (E) its isometry group. To avoid the obvious constructions mentioned above, we only consider cyclic self-repre- sentations (in the affine sense). It is well-known that there is a whole wealth of such self-representations. They are described by functions of conditionally nega- tive type. More precisely, the question becomes equivalent to describing all radial functions of conditionally negative type on E because one can arrange, by con- arXiv:1805.12479v1 [math.GR] 31 May 2018 jugating, that O(E) maps to itself. Thus the question reduces to the study of a fascinating space of functions Ψ: R+ R+, and new such functions can be obtained by composing a given Ψ with any→ Bernstein function [16]. We see that this first example, Is(E), has many — almost too many — self- representations for a precise classification. How about other infinite-dimensional groups? Are they too rigid to admit any, or again so soft as to admit too many? Considering that Is(E) sits in the much larger Möbius group M¨ob(E) of E, this article answers the following questions: Date: May 2018. ∗Partially supported by the french project ANR AGIRA and by project papiit IA100917 from DGAPA UNAM. 2 NICOLASMONODANDPIERREPY Does any non-tautological Is(E)-representation extend to M¨ob(E)? • Can the irreducible self-representations of M¨ob(E) be classified? • Among all Bernstein functions, which ones correspond to Möbius repre- • sentations? In short, the answer is that the situation is much more rigid than for the isome- tries Is(E), but still remains much richer than in the finite-dimensional case. Specifically, there is exactly a one-parameter family of self-representations. This appears as a continuous deformation of the tautological representation, given by the Bernstein functions x xt where the parameter t ranges in the interval (0, 1]. 7→ 1.B. Formal statements. Recall that the Möbius group M¨ob(E) is a group of transformations of the conformal sphere E = E ; it is generated by the isometries of E, which fix , and by the inversions∪v {∞}(r/ v )2v, where r> 0 b is the inversion radius (see∞ e.g. [14, I.3]). In particular7→ it containsk k all homotheties. A first basic formalisation of the existence part of our results is as follows. Let E be an infinite-dimensional separable real Hilbert space. Theorem I. For every 0 < t 1 there exists a continuous self-representation M¨ob(E) M¨ob(E) with the following≤ properties: → it restricts to an affinely irreducible self-representation of Is(E), • it maps the translation by v E to an isometry with translation part of • norm v t, ∈ it mapsk homothetiesk of dilation factor λ to similarities of dilation factor λt. • Before turning to our classification theorem, we recast the discussion into a more suggestive geometric context by switching to the viewpoint of Lobachevsky spaces. Recall that there exists three space forms: Euclidean, spherical and real hyperbolic. In infinite dimensions, this corresponds to Hilbert spaces, Hilbert spheres and the infinite-dimensional real hyperbolic space H∞. (For definiteness, we take all our spaces to be separable in this introduction.) The hyperbolic Riemannian metric of curvature 1 induces a distance d on H∞ and we consider the corresponding (Polish) isometry− group Is(H∞), which also acts on the boundary at infinity ∂H∞. After choosing a point in ∂H∞, we can, just like in the finite-dimensional case, identify ∂H∞ with the sphere E in a way that induces an isomorphism b H∞ Is( ) ∼= M¨ob(E). As will be recalled in Section 2, there is a linear model for Is(H∞), and hence the usual notion of irreducibility — which happens to coincide with a natural geomet- ric notion. Here is the geometric, and more precise, counterpart to Theorem I. Theorem Ibis. For every 0 < t 1 there exists a continuous irreducible self- representation ≤ ̺∞ : Is(H∞) Is(H∞) t −→ SELF-REPRESENTATIONSOFTHEMÖBIUSGROUP 3 ∞ and a ̺t -equivariant embedding f ∞ : H∞ H∞ t −→ such that for all x,y H∞ we have ∈ ∞ ∞ t (1.i) cosh d ft (x),ft (y) = cosh d(x,y) . ∞ Intuitively, the representations ̺t are limiting objects for the representations ̺n : Is(Hn) Is(H∞) t −→ of the finite-dimensional Is(Hn) that we studied in [13], although the latter do not have a simple explicit expression like (1.i) for the distance. Using this relation n ∞ to ̺t and our earlier results, we can show that these ̺t , which are unique up to conjugacy for each t, exhaust all possible irreducible self-representations. Theorem II. Every irreducible self-representation of Is(H∞) is conjugated to a unique representation ̺∞ as in Theorem Ibis for some 0 <t 1. t ≤ ∞ Even though we stated above that ̺t is, in some sense, a limit of representa- n Hn tions ̺t for Is( ) as n goes to infinity, there is a priori a difficulty in making this precise. Consider indeed a standard nested sequence of totally geodesic subspaces Hn Hn+1 . Hm . H∞ ⊆ ⊆ ⊆ ⊆ ⊆ m Hn with dense union. For m n, we can restrict ̺t to a copy of Is( ). After ≥ n passing to the irreducible component, this creates a copy of ̺t . For each new m n, there is a new positive definite component that needs to be taken into ≥ m Hm H∞ account. Likewise, the associated harmonic map ft : from [13], Hn n → when restricted to , does not coincide with ft and indeed is not harmonic on Hn n ; rather, it lies at finite distance from ft . Our strategy for eliminatingallthese difficulties is not to work with the groups, and not to work with the spaces either. Instead, we only keep track of the various distance functions and study their pointwise convergence. We then show that the spaces and groups can be reconstructed from this data after this easier limit has been established. This limiting behaviour of the distance functions is a basic instance of the phenomenon of concentration of measure on spheres. This strategyisvery much thesame astheone behindtheuse of kernels of pos- itive type to construct orthogonal representations, and of kernels of conditionally negative type to construct affine isometric actions. Specifically, we use the notion of kernels of hyperbolic type and establish the necessary reconstruction results. 1.C. Further considerations. The reader will have noticed that there is no con- tinuity assumption in Theorem II, although the representations constructed in Theorem Ibis are continuous. Indeed this formulation of Theorem II necessitates the following result, which leverages the automatic continuity proved by Tsankov in [17] for the orthogonal group. Theorem III. Every irreducible self-representation of Is(H∞) is continuous. 4 NICOLASMONODANDPIERREPY Such an automatic continuity phenomenon can fail in finite dimensions due to the fact that algebraic groups admit discontinuous endomorphisms induced by wild field endomorphisms (see Lebesgue [10, p. 533] and [9]). We do not know whether Is(H∞) enjoys the full strength of the automatic continuity established by Tsankov for the orthogonal group. Problem. Is every homomorphism from Is(H∞) to any separable topological group continuous? In another direction, the existence of interesting self-representation of Is(H∞) raises the possibility of composing them. The geometric description of Theo- bis ∞ ∞ ∞ rem I suggests that the composition of ̺t with ̺s should be related to ̺ts . ∞ ∞ Indeed, the considerations of Section 5.A show readily that ̺t ̺s remains non- elementary in the sense recalled below and hence admits a unique◦ irreducible ∞ sub-representation necessarily isomorphic to ̺ts . In other words, Theorem II implies that upon co-restricting to the unique irre- ducible part, the semi-group of irreducible self-representations modulo conjuga- tion is isomorphic to the multiplicative semi-group (0, 1]. We could not ascertain that it is really necessary to extract the irreducible part. Problem. Is the composition of two irreducible self-representations of Is(H∞) still irreducible? The last section of this article adds a few elements to the study in [13] of the n Hn representations ̺t of Is( ) for n finite. Contents 1. Introduction 1 2. Notation and preliminaries 4 3. Kernels of hyperbolic type 6 4. On representations arising from kernels 11 5. Self-representations of Is(H∞) 13 6. Automatic continuity 17 7. Finite-dimensional post-scripta 18 References 20 2. Notation and preliminaries 2.A. Minkowski spaces. We work throughout over the field R of real numbers. The scalar products and norms of the various Hilbert spaces introduced below will all be denoted by , and .